A 20% ethanol solution R is mixed with ethanol solution S in the propo

Lots of info in this one, so I'll start by jotting down the facts and getting organized. Here's what my scratch paper looks like:

R = 20% eth
S = ???
T = 20% eth

1:3 R:S ---> 1:1 new solution:T ---> 31.25% eth

My very first though is that S has to have significantly more than 20% ethanol. Even though it's being diluted quite a bit, the resulting solution still has a lot more than 20% ethanol, so S must have been a pretty strong solution. But unfortunately, the answer choices are quite close to each other, so I don't think I can estimate - we'll have to do some reasoning.

If I just dive into the math, there's going to be a lot of calculating. Let's see if we can simplify things. The 'easiest' fact they gave me is that the new solution is mixed 1:1 with solution T. That is, the resulting solution, the one with 31.25% ethanol, is composed of 50% solution T, and 50% of the mystery solution (which itself is made of R and S).

Since it's 50/50 between T and the mystery R:S solution, I know that its ethanol concentration has to be halfway between the concentrations of those two solutions. (That's a general rule about mixtures: if you mix two things equally, the strength of the result has to be halfway between the strength of the original two substances. That's why you could mix whole milk with skim milk and basically get 2% milk. )

So, 31.25% is halfway between T (which is 20%) and the mystery solution.

20% --------- 31.25% ------- ???

Because 31.25% is 11.25 above 20, the mystery solution must be 11.25 above that. 31.25% + 11.25% = 42.5%. In other words, 31.25% is halfway between 20% and 42.5%.

Great! We just figured out that the combination of R and S has a concentration of 42.5%. That's a good first step.

Now what? Well, use some ideas about mixtures again. R and S were mixed in a ratio of 1:3. That is, there's 3 times as much S in the R:S mixture as there is R.

So, if we plot the different concentrations on a line, the concentration of the mixture should be three times closer to S's concentration as it is to R's concentration.

R=20% ----------------- combination = 42.5% ---- S = ???

The combination's concentration, 42.5%, is 22.5% away from R's concentration. Divide this by 3 to find out how far it should be from S's concentration. 22.5/3 = 7.5. Therefore, S's concentration is 7.5 greater than the combination's concentration. S's concentration is 42.5% + 7.5% = 50%.

The correct answer is C.